December  2021, 20(12): 4177-4193. doi: 10.3934/cpaa.2021152

Nondegeneracy of solutions for a class of cooperative systems on $ \mathbb{R}^n $

1. 

The City University of New York, CSI, Mathematics Department, Staten Island, New York 10314, USA

2. 

University of Cologne, Dept. of Mathematics & Computer Science, 50931 Cologne, Germany

* Corresponding author

Received  May 2021 Published  December 2021 Early access  September 2021

Fund Project: The first author was supported by the Alexander von Humboldt foundation and by MINECO grant MTM2017-84214-C2-1-P

We consider fully coupled cooperative systems on $ \mathbb{R}^n $ with coefficients that decay exponentially at infinity. Expanding some results obtained previously on bounded domain, we prove that the existence of a strictly positive supersolution ensures the first eigenvalue to exist and to be nonzero. This result is applied to show that the topological solutions for a Chern-Simons model, described by a semilinear system on $ \mathbb{R}^2 $ with exponential nonlinearity, are nondegenerate.

Citation: Marcello Lucia, Guido Sweers. Nondegeneracy of solutions for a class of cooperative systems on $ \mathbb{R}^n $. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4177-4193. doi: 10.3934/cpaa.2021152
References:
[1]

R. A. Adams, Compact imbeddings of weighted Sobolev spaces on unbounded domains, J. Differ. Equ., 9 (1971), 325-334.  doi: 10.1016/0022-0396(71)90085-4.

[2]

I. BirindelliE. Mitidieri and G. Sweers, Existence of the principal eigenvalue for cooperative elliptic systems in a general domain, Differ. Equ., 35 (1999), 326-334. 

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[4]

H. Brezis and F. Merle, Uniform estimates and blow up behavior for solutions of $- \Delta u = V(x)e^u$ in two dimensions, Commun. Partial Differ. Equ., 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797.

[5]

K. J. BrownC. Cosner and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on$\mathbb{R}^{n}$, Proc. Amer. Math. Soc., 109 (1990), 147-155.  doi: 10.2307/2048374.

[6]

J. Busca and B. Sirakov, Symmetry results for semi-linear elliptic systems in the whole space, J. Differ. Equ., 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.

[7]

L. Cardoulis, Principal eigenvalues for systems of Schrödinger equations defined in the whole space with indefinite weights, Math. Slovaca, 65 (2015), 1079-1094.  doi: 10.1515/ms-2015-0074.

[8]

K. C. Chang, Principal eigenvalue for weight matrix in elliptic systems, Nonlinear Anal., 46 (2001), 419-433.  doi: 10.1016/S0362-546X(00)00140-1.

[9]

S. S. Chern and J. Simons, Characteristic forms and geometric invariants, Ann. Math., 99 (1974), 48-69.  doi: 10.2307/1971013.

[10]

J. L. ChernZ. Y. Chen and C. S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles, Commun. Math. Phys., 296 (2010), 323-351.  doi: 10.1007/s00220-010-1021-z.

[11]

K. ChoeN. KimY. Lee and C. S. Lin, Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in$\mathbb{R} ^{2}$, J. Funct. Anal., 273 (2017), 1734-1761.  doi: 10.1016/j.jfa.2017.05.012.

[12]

J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, Integ. Equ. Operat. Theory, 2 (1979), 174-198.  doi: 10.1007/BF01682733.

[13]

D. G. de Figueiredo and E. Mitidieri, Maximum principles for cooperative elliptic systems, C. R. Acad. Sci. Paris Sér. I Math., 310 (1990), 49-52. 

[14]

G. Dunne, Mass degeneracies in self-dual models, Phys. Lett. B, 345 (1995), 452-457.  doi: 10.1016/0370-2693(94)01649-W.

[15]

X. Han and G. Huang, Existence theorems for a general $ 2\times 2$ non-Abelian Chern-Simons-Higgs system over a torus, J. Differ. Equ., 263 (2017), 1522-1551.  doi: 10.1016/j.jde.2017.03.017.

[16]

P. Hess and S. Senn, Another approach to elliptic eigenvalue problems with respect to indefinite weight functions,, in Nonlinear Analysis and Optimization, Springer, Berlin, 1984. doi: 10.1007/BFb0101496.

[17]

R. Janssen, The dirichlet problem for second order elliptic operators on unbounded domains, Appl. Anal., 19 (1985), 201-216.  doi: 10.1080/00036818508839545.

[18]

C. KimC. LeeP. KoB. H. Lee and H. Min, Schrödinger fields on the plane with $[U(1)]^N$ Chern-Simons interactions and generalized self-dual solitons, Phys. Rev. D, 48 (1993), 1821-1840.  doi: 10.1103/PhysRevD.48.1821.

[19]

C. S. LinA. C. Ponce and Y. Yang, A system of elliptic equations arising in Chern-Simons field theory, J. Funct. Anal., 247 (2007), 289-350.  doi: 10.1016/j.jfa.2007.03.010.

[20]

C. S. Lin and J. V. Prajapat, Vortex condensates for relativistic abelian Chern-Simons model with two Higgs scalar fields and two gauge fields on a torus, Comm. Math. Phys., 288 (2009), 311-347.  doi: 10.1007/s00220-009-0774-8.

[21]

E. Mitidieri and G. Sweers, Weakly coupled elliptic systems and positivity, Math. Nachr., 173 (1995), 259-286.  doi: 10.1002/mana.19951730115.

[22]

B. de Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149-153.  doi: 10.1007/BF01162028.

[23] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. 
[24] R. G. Pinsky, Positive Harmonic Functions and Diffusion,, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511526244.
[25]

M. H. Protter and H. F. Weinberger, On the spectrum of general second order operators, Bull. Amer. Math. Soc., 72 (1966), 251-255.  doi: 10.1090/S0002-9904-1966-11485-4.

[26]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, , Prentice-Hall, Inc., Englewood Cliffs, N. J. 1967

[27]

B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526.  doi: 10.1090/S0273-0979-1982-15041-8.

[28]

H. H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, Band 215. Springer-Verlag, New York-Heidelberg, 1974.

[29]

G. Sweers, Strong positivity in $C(\bar{\Omega})$ for elliptic systems, Math. Z., 209 (1992), 251-271.  doi: 10.1007/BF02570833.

[30]

G. Tarantello, Uniqueness of selfdual periodic Chern-Simons vortices of topological-type, Calc. Var. Partial Differ. Equ., 29 (2007), 191-217.  doi: 10.1007/s00526-006-0062-9.

[31]

Y. Yang, The relativistic non-abelian Chern-Simons equations, Commun. Math. Phys., 186 (1997), 199-218.  doi: 10.1007/BF02885678.

show all references

References:
[1]

R. A. Adams, Compact imbeddings of weighted Sobolev spaces on unbounded domains, J. Differ. Equ., 9 (1971), 325-334.  doi: 10.1016/0022-0396(71)90085-4.

[2]

I. BirindelliE. Mitidieri and G. Sweers, Existence of the principal eigenvalue for cooperative elliptic systems in a general domain, Differ. Equ., 35 (1999), 326-334. 

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[4]

H. Brezis and F. Merle, Uniform estimates and blow up behavior for solutions of $- \Delta u = V(x)e^u$ in two dimensions, Commun. Partial Differ. Equ., 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797.

[5]

K. J. BrownC. Cosner and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on$\mathbb{R}^{n}$, Proc. Amer. Math. Soc., 109 (1990), 147-155.  doi: 10.2307/2048374.

[6]

J. Busca and B. Sirakov, Symmetry results for semi-linear elliptic systems in the whole space, J. Differ. Equ., 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.

[7]

L. Cardoulis, Principal eigenvalues for systems of Schrödinger equations defined in the whole space with indefinite weights, Math. Slovaca, 65 (2015), 1079-1094.  doi: 10.1515/ms-2015-0074.

[8]

K. C. Chang, Principal eigenvalue for weight matrix in elliptic systems, Nonlinear Anal., 46 (2001), 419-433.  doi: 10.1016/S0362-546X(00)00140-1.

[9]

S. S. Chern and J. Simons, Characteristic forms and geometric invariants, Ann. Math., 99 (1974), 48-69.  doi: 10.2307/1971013.

[10]

J. L. ChernZ. Y. Chen and C. S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles, Commun. Math. Phys., 296 (2010), 323-351.  doi: 10.1007/s00220-010-1021-z.

[11]

K. ChoeN. KimY. Lee and C. S. Lin, Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in$\mathbb{R} ^{2}$, J. Funct. Anal., 273 (2017), 1734-1761.  doi: 10.1016/j.jfa.2017.05.012.

[12]

J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, Integ. Equ. Operat. Theory, 2 (1979), 174-198.  doi: 10.1007/BF01682733.

[13]

D. G. de Figueiredo and E. Mitidieri, Maximum principles for cooperative elliptic systems, C. R. Acad. Sci. Paris Sér. I Math., 310 (1990), 49-52. 

[14]

G. Dunne, Mass degeneracies in self-dual models, Phys. Lett. B, 345 (1995), 452-457.  doi: 10.1016/0370-2693(94)01649-W.

[15]

X. Han and G. Huang, Existence theorems for a general $ 2\times 2$ non-Abelian Chern-Simons-Higgs system over a torus, J. Differ. Equ., 263 (2017), 1522-1551.  doi: 10.1016/j.jde.2017.03.017.

[16]

P. Hess and S. Senn, Another approach to elliptic eigenvalue problems with respect to indefinite weight functions,, in Nonlinear Analysis and Optimization, Springer, Berlin, 1984. doi: 10.1007/BFb0101496.

[17]

R. Janssen, The dirichlet problem for second order elliptic operators on unbounded domains, Appl. Anal., 19 (1985), 201-216.  doi: 10.1080/00036818508839545.

[18]

C. KimC. LeeP. KoB. H. Lee and H. Min, Schrödinger fields on the plane with $[U(1)]^N$ Chern-Simons interactions and generalized self-dual solitons, Phys. Rev. D, 48 (1993), 1821-1840.  doi: 10.1103/PhysRevD.48.1821.

[19]

C. S. LinA. C. Ponce and Y. Yang, A system of elliptic equations arising in Chern-Simons field theory, J. Funct. Anal., 247 (2007), 289-350.  doi: 10.1016/j.jfa.2007.03.010.

[20]

C. S. Lin and J. V. Prajapat, Vortex condensates for relativistic abelian Chern-Simons model with two Higgs scalar fields and two gauge fields on a torus, Comm. Math. Phys., 288 (2009), 311-347.  doi: 10.1007/s00220-009-0774-8.

[21]

E. Mitidieri and G. Sweers, Weakly coupled elliptic systems and positivity, Math. Nachr., 173 (1995), 259-286.  doi: 10.1002/mana.19951730115.

[22]

B. de Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149-153.  doi: 10.1007/BF01162028.

[23] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. 
[24] R. G. Pinsky, Positive Harmonic Functions and Diffusion,, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511526244.
[25]

M. H. Protter and H. F. Weinberger, On the spectrum of general second order operators, Bull. Amer. Math. Soc., 72 (1966), 251-255.  doi: 10.1090/S0002-9904-1966-11485-4.

[26]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, , Prentice-Hall, Inc., Englewood Cliffs, N. J. 1967

[27]

B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526.  doi: 10.1090/S0273-0979-1982-15041-8.

[28]

H. H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, Band 215. Springer-Verlag, New York-Heidelberg, 1974.

[29]

G. Sweers, Strong positivity in $C(\bar{\Omega})$ for elliptic systems, Math. Z., 209 (1992), 251-271.  doi: 10.1007/BF02570833.

[30]

G. Tarantello, Uniqueness of selfdual periodic Chern-Simons vortices of topological-type, Calc. Var. Partial Differ. Equ., 29 (2007), 191-217.  doi: 10.1007/s00526-006-0062-9.

[31]

Y. Yang, The relativistic non-abelian Chern-Simons equations, Commun. Math. Phys., 186 (1997), 199-218.  doi: 10.1007/BF02885678.

Figure 1.  Typical behaviour of the potential $ x\mapsto V_{i}\left( x\right) $ in $ L_{i} $
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